Stochastic Differential Equation

A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, leading to a solution that is itself a stochastic process. SDEs are used to model systems that are influenced by random noise or uncertainty, such as financial markets, physical systems subject to thermal fluctuations, and biological processes. The formulation originates from Brownian Motion and Ito Calculus.

They consist of both a deterministic component and a stochastic component. The deterministic part describes the average behavior of the system, while the stochastic part accounts for random fluctuations.

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General Form

A typical SDE can be written in the form:

$$\frac{d{X}_t}{dt}=a(X_t,t)+b(X_t,t)\cdot \frac{d{W}_t}{dt}$$

Where:

• $a(X_t,t)$ is the drift term, representing the deterministic part of the equation.
• $b(X_t,t)$ is the diffusion term, representing the intensity of the stochastic
• $\frac{d{W}_t}{dt}$ is the derivative of a Wiener process (or Brownian motion), representing the random noise.

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Python Implementation

This example simulates multiple paths of an Ornstein-Uhlenbeck process, a common type of SDE used in various fields such as finance and physics.¹ The Ornstein-Uhlenbeck process is defined by the SDE:

Python

import numpy as np import matplotlib.pyplot as plt # Parameters for the Ornstein-Uhlenbeck process # dX(t) = theta * (mu - X(t)) * dt + sigma * dW(t) theta = 2.0 # mean reversion speed mu = 0.0 # long-term mean sigma = 1.0 # volatility X0 = 2.0 # initial value # Simulation parameters T = 2.0 # total time N = 1000 # number of time steps dt = T / N # time step size n_paths = 10 # number of paths to simulate # Time array t = np.linspace(0, T, N + 1) # Initialize array to store all paths paths = np.zeros((n_paths, N + 1)) paths[:, 0] = X0 # set initial condition for all paths # Generate random increments np.random.seed(42) # for reproducible results dW = np.random.normal(0, np.sqrt(dt), (n_paths, N)) # Simulate the SDE using Euler-Maruyama method for i in range(N): for path in range(n_paths): drift = theta * (mu - paths[path, i]) * dt diffusion = sigma * dW[path, i] paths[path, i + 1] = paths[path, i] + drift + diffusion # Create the plot plt.figure(figsize=(12, 8)) # Plot all paths colors = plt.cm.tab10(np.linspace(0, 1, n_paths)) for i in range(n_paths): plt.plot(t, paths[i, :], color=colors[i], alpha=0.7, linewidth=1.5, label=f'Path {i+1}' if i < 5 else "") # Only show first 5 in legend # Add theoretical mean theoretical_mean = mu + (X0 - mu) * np.exp(-theta * t) plt.plot(t, theoretical_mean, 'k--', linewidth=2, alpha=0.8, label='Theoretical Mean') # Formatting plt.xlabel('Time (t)', fontsize=12) plt.ylabel('X(t)', fontsize=12) plt.title(f'Ornstein-Uhlenbeck Process: {n_paths} Stochastic Paths\n' + f'dX(t) = {theta}(μ - X(t))dt + {sigma}dW(t), X(0) = {X0}', fontsize=14) plt.grid(True, alpha=0.3) plt.legend(loc='upper right') # Add statistics text box mean_final = np.mean(paths[:, -1]) std_final = np.std(paths[:, -1]) textstr = f'Final values:\nMean: {mean_final:.3f}\nStd: {std_final:.3f}\nTheoretical mean: {mu:.3f}' props = dict(boxstyle='round', facecolor='wheat', alpha=0.5) plt.text(0.02, 0.98, textstr, transform=plt.gca().transAxes, fontsize=10, verticalalignment='top', bbox=props) plt.tight_layout() plt.show() # Print some statistics print(f"Simulation completed with {n_paths} paths over time [0, {T}]") print(f"Initial value: {X0}") print(f"Final mean: {mean_final:.4f}") print(f"Final standard deviation: {std_final:.4f}") print(f"Theoretical long-term mean: {mu}") print(f"Mean reversion parameter θ: {theta}") print(f"Volatility σ: {sigma}")

Output

Footnotes

1. https://en.wikipedia.org/wiki/Stochastic_differential_equation ↩